Smoothness of the attractor of almost all solutions of a delay differential equation

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AbstractLet a C¹-function f:ℝ → ℝ be given which satisfies f(0) = 0, f'(ξ) < 0 for all ξ ∈ ℝ, and sup f < ∞ or -∞ < inf f. Let C = C([-1,0],ℝ). For an open-dense set of initial data the phase curves [0,∞) → C given by the solutions [-1,∞) → ℝ to the negative feedback equationx'(t) = -μx(t) + f(x(t-1)), with μ > 0,are absorbed into the positively invariant set S ⊂ C of data ϕ ≠ 0 with at most one sign change. The global attractor A of the semiflow restricted to S̅ is either the singleton {0} or it is given by a Lipschitz continuous map a with domain pA in a 2-dimensional subspace L ⊂ C and range in a complementary subspace Q; pA is homeomorphic to the closed unit disk in ℝ². We show that a is in fact C¹-smooth.CONTENTS1. Introduction......................................................................................................................................................................52. The delay differential equation and its attractor of almost all solutions............................................................................9 2.1. The delay differential equation....................................................................................................................................9 2.2. Slowly oscillating solutions.........................................................................................................................................13 2.3. The attractor of eventually slowly oscillating solutions...............................................................................................16 2.4. Floquet multipliers of slowly oscillating periodic solutions and adapted Poincaré maps.............................................21 2.5. Local invariant manifolds...........................................................................................................................................273. A-priori estimates...........................................................................................................................................................33 3.1. Nonautonomous equations........................................................................................................................................33 3.2. Vectors tangent to the attractor and to domains of adapted Poincaré maps.............................................................394. Transversals on the attractor and smoothness..............................................................................................................41 4.1. A sufficient condition for smoothness.........................................................................................................................41 4.2. Smoothness at wandering points...............................................................................................................................425. Curves on the attractor emanating from periodic orbits and connecting the stationary point to a periodic orbit............44 5.1. From lines in the plane L to curves on the graph A which are transversal to the flow................................................44 5.2. Arcs emanating from periodic orbits..........................................................................................................................45 5.3. Smooth ends at periodic orbits..................................................................................................................................47 5.4. A curve on A connecting 0 in K̅ to a periodic orbit.....................................................................................................536. Smoothness at periodic orbits.......................................................................................................................................59 6.1. Interior periodic orbits................................................................................................................................................59 6.2. Smoothness at the boundary.....................................................................................................................................627. Smoothness at the stationary point................................................................................................................................63 7.1. Cases of no attraction................................................................................................................................................63 7.2. On the inclination of tangent spaces of the attractor close to the stationary point.....................................................65 7.3. The cases of attraction..............................................................................................................................................68References........................................................................................................................................................................711991 Mathematics Subject Classification: 34K15, 58F12.

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Walther Hans-Otto, and Yebdri Mustapha. Smoothness of the attractor of almost all solutions of a delay differential equation. Warszawa: Instytut Matematyczny Polskiej Akademi Nauk, 1997. <http://eudml.org/doc/271130>.

@book{WaltherHans1997, abstract = {AbstractLet a C¹-function f:ℝ → ℝ be given which satisfies f(0) = 0, f'(ξ) < 0 for all ξ ∈ ℝ, and sup f < ∞ or -∞ < inf f. Let C = C([-1,0],ℝ). For an open-dense set of initial data the phase curves [0,∞) → C given by the solutions [-1,∞) → ℝ to the negative feedback equationx'(t) = -μx(t) + f(x(t-1)), with μ > 0,are absorbed into the positively invariant set S ⊂ C of data ϕ ≠ 0 with at most one sign change. The global attractor A of the semiflow restricted to S̅ is either the singleton \{0\} or it is given by a Lipschitz continuous map a with domain pA in a 2-dimensional subspace L ⊂ C and range in a complementary subspace Q; pA is homeomorphic to the closed unit disk in ℝ². We show that a is in fact C¹-smooth.CONTENTS1. Introduction......................................................................................................................................................................52. The delay differential equation and its attractor of almost all solutions............................................................................9 2.1. The delay differential equation....................................................................................................................................9 2.2. Slowly oscillating solutions.........................................................................................................................................13 2.3. The attractor of eventually slowly oscillating solutions...............................................................................................16 2.4. Floquet multipliers of slowly oscillating periodic solutions and adapted Poincaré maps.............................................21 2.5. Local invariant manifolds...........................................................................................................................................273. A-priori estimates...........................................................................................................................................................33 3.1. Nonautonomous equations........................................................................................................................................33 3.2. Vectors tangent to the attractor and to domains of adapted Poincaré maps.............................................................394. Transversals on the attractor and smoothness..............................................................................................................41 4.1. A sufficient condition for smoothness.........................................................................................................................41 4.2. Smoothness at wandering points...............................................................................................................................425. Curves on the attractor emanating from periodic orbits and connecting the stationary point to a periodic orbit............44 5.1. From lines in the plane L to curves on the graph A which are transversal to the flow................................................44 5.2. Arcs emanating from periodic orbits..........................................................................................................................45 5.3. Smooth ends at periodic orbits..................................................................................................................................47 5.4. A curve on A connecting 0 in K̅ to a periodic orbit.....................................................................................................536. Smoothness at periodic orbits.......................................................................................................................................59 6.1. Interior periodic orbits................................................................................................................................................59 6.2. Smoothness at the boundary.....................................................................................................................................627. Smoothness at the stationary point................................................................................................................................63 7.1. Cases of no attraction................................................................................................................................................63 7.2. On the inclination of tangent spaces of the attractor close to the stationary point.....................................................65 7.3. The cases of attraction..............................................................................................................................................68References........................................................................................................................................................................711991 Mathematics Subject Classification: 34K15, 58F12.}, author = {Walther Hans-Otto, Yebdri Mustapha}, keywords = {delay differential equation; negative feedback equation; global attractor; semiflow}, language = {eng}, location = {Warszawa}, publisher = {Instytut Matematyczny Polskiej Akademi Nauk}, title = {Smoothness of the attractor of almost all solutions of a delay differential equation}, url = {http://eudml.org/doc/271130}, year = {1997},}

TY - BOOKAU - Walther Hans-OttoAU - Yebdri MustaphaTI - Smoothness of the attractor of almost all solutions of a delay differential equationPY - 1997CY - WarszawaPB - Instytut Matematyczny Polskiej Akademi NaukAB - AbstractLet a C¹-function f:ℝ → ℝ be given which satisfies f(0) = 0, f'(ξ) < 0 for all ξ ∈ ℝ, and sup f < ∞ or -∞ < inf f. Let C = C([-1,0],ℝ). For an open-dense set of initial data the phase curves [0,∞) → C given by the solutions [-1,∞) → ℝ to the negative feedback equationx'(t) = -μx(t) + f(x(t-1)), with μ > 0,are absorbed into the positively invariant set S ⊂ C of data ϕ ≠ 0 with at most one sign change. The global attractor A of the semiflow restricted to S̅ is either the singleton {0} or it is given by a Lipschitz continuous map a with domain pA in a 2-dimensional subspace L ⊂ C and range in a complementary subspace Q; pA is homeomorphic to the closed unit disk in ℝ². We show that a is in fact C¹-smooth.CONTENTS1. Introduction......................................................................................................................................................................52. The delay differential equation and its attractor of almost all solutions............................................................................9 2.1. The delay differential equation....................................................................................................................................9 2.2. Slowly oscillating solutions.........................................................................................................................................13 2.3. The attractor of eventually slowly oscillating solutions...............................................................................................16 2.4. Floquet multipliers of slowly oscillating periodic solutions and adapted Poincaré maps.............................................21 2.5. Local invariant manifolds...........................................................................................................................................273. A-priori estimates...........................................................................................................................................................33 3.1. Nonautonomous equations........................................................................................................................................33 3.2. Vectors tangent to the attractor and to domains of adapted Poincaré maps.............................................................394. Transversals on the attractor and smoothness..............................................................................................................41 4.1. A sufficient condition for smoothness.........................................................................................................................41 4.2. Smoothness at wandering points...............................................................................................................................425. Curves on the attractor emanating from periodic orbits and connecting the stationary point to a periodic orbit............44 5.1. From lines in the plane L to curves on the graph A which are transversal to the flow................................................44 5.2. Arcs emanating from periodic orbits..........................................................................................................................45 5.3. Smooth ends at periodic orbits..................................................................................................................................47 5.4. A curve on A connecting 0 in K̅ to a periodic orbit.....................................................................................................536. Smoothness at periodic orbits.......................................................................................................................................59 6.1. Interior periodic orbits................................................................................................................................................59 6.2. Smoothness at the boundary.....................................................................................................................................627. Smoothness at the stationary point................................................................................................................................63 7.1. Cases of no attraction................................................................................................................................................63 7.2. On the inclination of tangent spaces of the attractor close to the stationary point.....................................................65 7.3. The cases of attraction..............................................................................................................................................68References........................................................................................................................................................................711991 Mathematics Subject Classification: 34K15, 58F12.LA - engKW - delay differential equation; negative feedback equation; global attractor; semiflowUR - http://eudml.org/doc/271130ER -